$A$ metallic solid sphere is rotating about its diameter as an axis of rotation. If the temperature is increased by $200^{\circ} C$, the percentage increase in its moment of inertia is (Coefficient of linear expansion of the metal $= 10^{-5} /^{\circ} C$) (in $\%$)

Two circular loops $P$ and $Q$ of radii $r$ and $nr$ are made respectively from a uniform wire. The moment of inertia of loop $Q$ about its axis is $4$ times that of loop $P$ about its axis. The value of $n$ is:

The radius of gyration of a rod of length $L$ and mass $M$ about an axis perpendicular to its length and passing through a point at a distance $L/3$ from one of its ends,is :-

Let $M$ and $L$ be the mass and length of a thin uniform rod,respectively. In the $1^{\text{st}}$ case,the axis of rotation passes through the center and is perpendicular to its length. In the $2^{\text{nd}}$ case,the axis of rotation passes through one end and is perpendicular to its length. The ratio of the radius of gyration in the first case to the second case is:

Three identical rods,each of length $l$ and mass $M$,are joined to form a rigid equilateral triangle. Its radius of gyration about an axis passing through a corner and perpendicular to the plane of the triangle is

$A$ straight rod of length $L$ is made of a material having mass per unit length $m(x) = \lambda|x|$, where $x$ is measured from the center of the rod. The moment of inertia about an axis perpendicular to the rod and passing through one end of the rod is to be calculated. Given $L = 1 \ m$ and $\lambda = 16 \ kg/m^2$.